Answer :
To solve this problem, we need to determine the interval of time during which Jerald is within 104 feet above the ground, based on the height equation: [tex]\( h = -16t^2 + 729 \)[/tex].
Here’s a step-by-step solution:
1. Understand the equation: The height [tex]\( h \)[/tex] in feet at any time [tex]\( t \)[/tex] in seconds is given by the equation [tex]\( h = -16t^2 + 729 \)[/tex]. The term [tex]\(-16t^2\)[/tex] indicates how the height changes over time due to gravity, with 729 being the initial height from which Jerald jumps.
2. Set up the inequality: We want to find the values of [tex]\( t \)[/tex] when Jerald is within 104 feet above the ground, meaning we want to solve the inequality:
[tex]\[
-16t^2 + 729 \leq 104
\][/tex]
3. Rearrange the inequality:
[tex]\[
-16t^2 + 729 - 104 \leq 0
\][/tex]
[tex]\[
-16t^2 + 625 \leq 0
\][/tex]
Divide the entire inequality by -16 (remembering to reverse the inequality sign when you divide or multiply by a negative number):
[tex]\[
t^2 \geq \frac{625}{16}
\][/tex]
4. Solve for [tex]\( t \)[/tex]:
[tex]\[
t^2 \geq 39.0625
\][/tex]
Taking the square root of both sides gives:
[tex]\[
|t| \geq \sqrt{39.0625}
\][/tex]
Therefore:
[tex]\[
t \leq -6.25 \quad \text{or} \quad t \geq 6.25
\][/tex]
5. Determine the feasible interval: Since [tex]\( t \)[/tex] represents time, it must be non-negative. Hence, we only consider the positive value:
[tex]\[
t \geq 6.25
\][/tex]
However, this tells us only when Jerald is below 104 feet. We want to find when he is above 104 feet, which is the opposite of what we found. So, within the interval when [tex]\( t = 0 \)[/tex] (as Jerald jumps) and [tex]\( t = 6.25 \)[/tex], he is above 104 feet. Thus, the interval when he is within 104 feet from the ground is:
[tex]\[ 0 \leq t \leq 6.25 \][/tex]
So the answer is c) [tex]\(0 \leq t \leq 6.25\)[/tex].
Here’s a step-by-step solution:
1. Understand the equation: The height [tex]\( h \)[/tex] in feet at any time [tex]\( t \)[/tex] in seconds is given by the equation [tex]\( h = -16t^2 + 729 \)[/tex]. The term [tex]\(-16t^2\)[/tex] indicates how the height changes over time due to gravity, with 729 being the initial height from which Jerald jumps.
2. Set up the inequality: We want to find the values of [tex]\( t \)[/tex] when Jerald is within 104 feet above the ground, meaning we want to solve the inequality:
[tex]\[
-16t^2 + 729 \leq 104
\][/tex]
3. Rearrange the inequality:
[tex]\[
-16t^2 + 729 - 104 \leq 0
\][/tex]
[tex]\[
-16t^2 + 625 \leq 0
\][/tex]
Divide the entire inequality by -16 (remembering to reverse the inequality sign when you divide or multiply by a negative number):
[tex]\[
t^2 \geq \frac{625}{16}
\][/tex]
4. Solve for [tex]\( t \)[/tex]:
[tex]\[
t^2 \geq 39.0625
\][/tex]
Taking the square root of both sides gives:
[tex]\[
|t| \geq \sqrt{39.0625}
\][/tex]
Therefore:
[tex]\[
t \leq -6.25 \quad \text{or} \quad t \geq 6.25
\][/tex]
5. Determine the feasible interval: Since [tex]\( t \)[/tex] represents time, it must be non-negative. Hence, we only consider the positive value:
[tex]\[
t \geq 6.25
\][/tex]
However, this tells us only when Jerald is below 104 feet. We want to find when he is above 104 feet, which is the opposite of what we found. So, within the interval when [tex]\( t = 0 \)[/tex] (as Jerald jumps) and [tex]\( t = 6.25 \)[/tex], he is above 104 feet. Thus, the interval when he is within 104 feet from the ground is:
[tex]\[ 0 \leq t \leq 6.25 \][/tex]
So the answer is c) [tex]\(0 \leq t \leq 6.25\)[/tex].
